Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as $\tau$-Functions -- Spectrum Singularity case

Abstract

The probability for the exclusion of eigenvalues from an interval $(-x,x)$ symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter $a$ (a generalisation of the sine kernel in the bulk scaling case), is considered. It is shown that this probability is the square of a $\tau$-function, in the sense of Okamoto, for the Painlevé system \PIII. This then leads to a factorisation of the probability as the product of two $\tau$-functions for the Painlevé system \PIIIdash. A previous study has given a formula of this type but involving \PIIIdash systems with different parameters consequently implying an identity between products of $\tau$-functions or equivalently sums of Hamiltonians.

AMSLatex, 9 pages

References in corpus (2)

• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
• {\bf $\tau$-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}

Cited by in corpus (1)

• Entanglement entropies of an interval in the free Schrödinger field theory on the half line